Optimal Propellant Ratios for Specific Impulse

Choosing the correct fuel-oxygen ratio depends on balancing maximum energetic performance (I_sp) with constraints on combustion temperature, materials limits, and propellant density. 

In space rocket engines, the terms stoichiometric and optimal refer to the ratio of oxidizer to fuel (O/F ratio) used during combustion. While a stoichiometric mixture provides the highest potential temperature, it is almost never used in rocket engines because it is not efficient in practice. Instead, engineers use an "optimal" mixture, which balances maximum energy, exhaust velocity, and material limits. 

Stoichiometric Ratio is the perfect chemical balance where there is exactly enough oxidizer to burn all the fuel completely, leaving no unreacted reactants. It produces maximum possible combustion temperature. However, the temperatures produced are usually too hot, which can melt the engine's turbine blades and combustion chamber walls. A stoichiometric mix usually creates heavier exhaust molecules (such as CO₂ instead of CO), resulting in lower specific impulse (I_sp). For fuels like hydrogen, stoichiometric mixture is much less dense, requiring larger, heavier tanks.

Optimal ratio produces the best overall system performance, specifically maximizing specific impulse (I_sp) - the efficiency of engine, or thrust produced per unit of propellant.  

Optimal ratios for common propellants refer to the ideal proportion of oxidizer to fuel (the mixture ratio or O/F ratio) that provides the maximum performance, usually defined as the highest specific impulse (I_sp) or the most efficient thrust, for a rocket engine. Stoichiometry ratio defines the ratio for complete combustion, the optimal ratio is often slightly different in order to maximize efficiency, reduce temperature, or optimize density. 

The mixture ratio is defined as the mass flow rate of the oxidizer (ṁ_o) divided by the mass flow rate of the fuel (ṁ_f): 

Mixture Ratio (O/F) = ṁ_o / ṁ_f

Optimal ratios depend on whether the goal is maximum energy, maximum density, or lower combustion temperatures. 

What it means Optimal?

• Maximum I_sp vs. Stoichiometric: The ratio that gives maximum I_sp (highest efficiency) is not always the stoichiometric ratio (complete combustion). Usually, running slightly "fuel-rich" (less oxidizer) is best because it reduces the average molecular weight of the exhaust gases, which boosts performance, even if the combustion temperature is slightly lower.
• Density vs. Performance (Density Impulse): In some cases, a higher O/F ratio is used to make the fuel denser. A denser propellant allows for smaller tanks, reducing structural weight. This is a tradeoff between higher I_sp and lower vehicle weight.
• Temperature Constraints: Running stoichiometric can result in temperatures too high for the nozzle/chamber materials. Running fuel-rich lowers the temperature, increasing engine life.
• Volume-Constrained Design: For hydrogen-fueled rockets, the fuel is very low density, requiring enormous tanks. Running at a higher O/F ratio makes the fuel denser, reducing tank size. 

Examples of fuel/oxidizers and their specific values: 

• LOX/LH₂ (Liquid Oxygen/Liquid Hydrogen): While the stoichiometric ratio is 8:1, the optimum for maximum I_sp is usually around 4:1 to 5:1. However, to improve density and reduce tank size, they are often run at 5.5:1 or higher (e.g., Space Shuttle Main Engine ran at 6:1).
• LOX/RP-1 (Liquid Oxygen/Kerosene): The ideal stoichiometric ratio for LOX/RP-1 is approximately 3.4:1, but practical rocket engines typically run fuel-rich, utilizing an oxidizer-to-fuel (O/F) mass ratio between 2.1:1 and 2.7:1 to maximize performance. A ratio of 2.56-2.58:1 is frequently used for maximum performance. 
• Operational Ratios:
• Merlin 1D (Falcon 9): Approximately 2.3:1.
• Saturn V (F-1 Engine): Approx. 2.27:1.
• Common Range: 2.4-2.6:1 (slightly fuel-rich for lower temperature and higher specific impulse). 
• LOX/Methane (Liquid Oxygen/Methane): The stoichiometric ratio for liquid oxygen and methane is approximately 3.6 to 4.0 by mass. This stoichiometric ratio of 1 kg of methane to roughly 3.6-4 kg of oxygen provides complete combustion. However, rocket engines (such as Raptor) often operate slightly fuel-rich to optimize performance and thermal management. The optimum mixture ratio is typically around 3.5 to improve engine performance. 

Starship ignition during launch; source: Wiki

• NTO/MMH (Nitrogen Tetroxide/Monomethylhydrazine): These hypergolic propellants are often used in spacecraft. The stoichiometric ratio is approximately 1.56 to 1.67. An O/F ratio of 1.67 is often chosen, not just for efficiency, but to make the tank volumes equal for easier packaging. Optimal ratio is typically around 1.6 to 1.65.
• While thermodynamic predictions might suggest a slightly higher ratio, actual industry practice for this combination favors lower ratios to maximize performance (often under the frozen flow assumption) and to manage engine temperature.
• Aerojet R-4D Family: Widely used for satellite attitude control, apogee insertion, and interplanetary probes.
• Aestus Engine: Used on the Ariane 5 upper stage.
• Space Shuttle OMS/RCS Engines: The Orbital Maneuvering System and Reaction Control System engines on the Space Shuttle used MMH and MON-3 (a variant of NTO).
• Apollo Service Propulsion System (SPS).

During the launch and moments after doesn't mean that O/F ratio remains the same. 

Stage 1 - Initial Thrust, while running on higher O/F: During liftoff, rockets require high thrust to overcome gravity. This often calls for a higher O/F ratio (more oxidizer relative to fuel) to produce higher thrust and maximize energy release.

Stage 2 - Efficiency Shift to lower O/F: As the rocket gets lighter and the ambient pressure decreases, the focus shifts to maximizing specific impulse (I_sp). Running fuel-rich (lower O/F) often increases by reducing the average molecular weight of the exhaust gases, allowing them to travel faster.

Stage 3 - Propellant Utilization: To avoid having significant leftover oxidizer or fuel when the other runs out, engines often switch to a different mixture ratio late in the burn to match consumption rates to the remaining tank capacities. Basically to get rid off whatever is remaining. 

In summary, the mixture ratio is a dynamic parameter, not a constant, designed to change from high-thrust ratios to high-efficiency ratios as the rocket ascends.